| L(s) = 1 | − 3-s + 0.699·5-s + 3.79·7-s + 9-s + 11-s − 13-s − 0.699·15-s − 5.09·17-s − 0.538·19-s − 3.79·21-s − 7.18·23-s − 4.51·25-s − 27-s − 9.14·29-s + 1.95·31-s − 33-s + 2.65·35-s − 1.39·37-s + 39-s − 1.79·41-s − 10.8·43-s + 0.699·45-s − 13.5·47-s + 7.37·49-s + 5.09·51-s + 9.90·53-s + 0.699·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.312·5-s + 1.43·7-s + 0.333·9-s + 0.301·11-s − 0.277·13-s − 0.180·15-s − 1.23·17-s − 0.123·19-s − 0.827·21-s − 1.49·23-s − 0.902·25-s − 0.192·27-s − 1.69·29-s + 0.350·31-s − 0.174·33-s + 0.448·35-s − 0.229·37-s + 0.160·39-s − 0.279·41-s − 1.65·43-s + 0.104·45-s − 1.98·47-s + 1.05·49-s + 0.712·51-s + 1.36·53-s + 0.0942·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 0.699T + 5T^{2} \) |
| 7 | \( 1 - 3.79T + 7T^{2} \) |
| 17 | \( 1 + 5.09T + 17T^{2} \) |
| 19 | \( 1 + 0.538T + 19T^{2} \) |
| 23 | \( 1 + 7.18T + 23T^{2} \) |
| 29 | \( 1 + 9.14T + 29T^{2} \) |
| 31 | \( 1 - 1.95T + 31T^{2} \) |
| 37 | \( 1 + 1.39T + 37T^{2} \) |
| 41 | \( 1 + 1.79T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + 13.5T + 47T^{2} \) |
| 53 | \( 1 - 9.90T + 53T^{2} \) |
| 59 | \( 1 - 5.72T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 + 9.90T + 71T^{2} \) |
| 73 | \( 1 - 3.83T + 73T^{2} \) |
| 79 | \( 1 - 1.37T + 79T^{2} \) |
| 83 | \( 1 + 4.32T + 83T^{2} \) |
| 89 | \( 1 - 5.95T + 89T^{2} \) |
| 97 | \( 1 - 1.95T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.222296425455849218835334177022, −7.50437563955240415824919454222, −6.68678829369983701451308265798, −5.90021256554390223137961940595, −5.18036215508076285460080728607, −4.47858158412144352559814110531, −3.73422464629083572793791279248, −2.10813196715114290463050909424, −1.67960676233759357872078128930, 0,
1.67960676233759357872078128930, 2.10813196715114290463050909424, 3.73422464629083572793791279248, 4.47858158412144352559814110531, 5.18036215508076285460080728607, 5.90021256554390223137961940595, 6.68678829369983701451308265798, 7.50437563955240415824919454222, 8.222296425455849218835334177022
